Planck's constant.

Dirac's canonical quantisation. (Identification of the quantum commutator with classical Poisson brackets).

Locality of quantum field theory.

The Born rule and definite outcomes of quantum measurements.

Could they emerge from some underlying theory?

Here is a brief summary of some of the key ideas in Quantum theory as an emergent phenomonen, by Stephen L. Adler. Helpful introductions are the slides [hand-written viewgraphs!] from a talk Adler gave in 2006 (see the summary slide below) and a review of the book by Philip Pearle. [There is also a draft of the book on the arXiv].

This is an incredibly original and creative proposal.

The starting point are "classical" dynamical variables pr, qr which are NxN matrices. Some bosonic and others are fermionic. They all obey Hamilton's equations of motion for an unspecified Hamiltonian H.

Three quantities are conserved, H, the fermion number N, and the traceless anti-self adjoint matrix:

where the first term is the sum of commutators over bosonic variables and the second the sum over anti-commutators over fermionic variables.

Quantum theory is obtained by tracing over all the classical variables with respect to an canonical ensemble with three (matrix) Lagrange multipliers corresponding to the conserved quantities H, N, and C.

The expectation value of the diagonal elements of C are assumed to all have the same value, hbar!

An analogy of the equipartition theorem (which looks like a Ward identity from field theory) leads to dynamical equations for effective fields. To make these equations look like quantum field theory an assumption is now made about a hierarchy of length, energy, and "temperature" [Lagrange mulitiplier] scales, which cause the Trace dynamics to be dominated by C rather than H, the trace Hamiltonian. Adler suggests these scales may be Planck scales. What then emerges are the usual quantum dynamical equations and the Dirac correspondence of Poisson brackets and commutators.

The "classical" field C fluctuates about its average value. These fluctuations can be identified with corrections to locality in quantum field theory and with the noise terms which appear in "physical collapse" models [e.g., Continuous Spontaneous Localisation] of quantum theory.

This is very impressive!

So some questions I have:

- How is the theory falsifiable? On some energy/time/length scales the underlying fluctuating classical fields must be experimentally accessible. But, will we ever be able to distinguish them from background decoherence?
- Is the assumption that all the expectation values of C have a simple diagonal form some version of a maximum entropy principle?
- This seems to be a "hidden variable theory" but with bosonic and fermionic hidden variables. How does this allow one to get around Bell's inequalities?
- The discussion of the fluctuating C field focuses on the part that causes collapse. [Im K(t)]. However, won't causality and Kramers-Kronig mean that this will have a real part. This will cause dynamical shifts in the energy eigenvalues of the Schrodinger equation? This could be easier to measure experimentally than collapse/decoherence. [See for example a discussion of dynamical Stokes shifts in biomolecules due interaction with their environment].

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